I was a assistant director of a nursing home 3rd in command, my company asked me to go to another?

nursing home for 6 weeks and help them out in a pinch, 80 miles from where I live, we get paid every 2 weeks on a Monday, I asked my boss if I could have my chek early on Friday when the usually arrive via mail, she said no I could not, I said i really needed the money before I left, she suggessted I have someone pick it up for me on monday, and then made a comment that i could get it after 12mn on sunday, so on Sun after 12mn I went in to work and went into the business office and got my check, (have a master key) since in my bossess absence I am in charge, I was fired for that they said It was breaking and entering in another persons office, what do you think, was I at fault

Do I think its fair not but do you have the right to go through the pay cheques?

If this is part of your normal duties its a grey area but if its not part of your duties its best avoided,

I am surprised they considered this breaking and entering as you have a key, do you normally have free access to the office?

I would seek advice from a qualified employment specialist and there appear to be a few areas that do not make sense.

One other point.

Do you think that they were hoping that by sending you 80 miles away you would have voted with your feet and left? It is possible that they wanted to get you out and just saw this as the perfect opportunity but breaking and entering seems extreme.

Consult with an employment attorney.
While you're waiting, what do the personnel policies say?
What policy did you violate?
It may have been poor judgment, but may or may not qualify as grounds for dismissal.
Many states are "employment at will" so either party can end the arrangement without any reason. So, even if they do not have grounds to dismiss you, you may not have much leverage.
Again, consult with an employment attorney.